The role of chaotic orbits in the determination of power spectra of passive scalars

Physics of Fluids

Volumn 8, Issue 11, 1996, Pages 3094-3104

  Antonsen Jr , Thomas M ^a   Fan, Zhencan ^a   Ott, Edward ^a   Garcia Lopez, E ^a  

^a UNIVERSITY OF MARYLAND   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000296895     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.869083     Document Type: Article

References (34)

1. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
2. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
3. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
4. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
5. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
6. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
7. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
8. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
9. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
10. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
11. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
12. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
13. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
14. Some recent works on the convection of passive scalars in flow fields are the following: H. Aref, "Stirring by chaotic advection," Fluid Mech. 143, 1 (1984); T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flow," ibid. 167, 353 (1986); J. Chaiken, C. K. Chu, M. Tabor, and Q. M. Tran, "Lagrangian turbulence and spatial complexity in a stochastic flow," Phys. Fluids 30, 687 (1987); D. V. Khakhar, H. Rising, and J. M. Ottino, "Analysis of chaotic mixing in two model systems," J. Fluid Mech. 172, 419 (1986); S. W. Jones and H. Aref, "Chaotic advection in pulsed source-sink systems," Phys. Fluids 31, 469 (1988); T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Bénard convection,́ ibid. 31, 1372 (1988); F. Heslot, B. Castaing, and A. Libshaber, "Transition to turbulence in helium gas," Phys. Rev. A 36, 5870 (1987); B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libshaber, S. Thomas, X. Z. Wu, S. Zaleski, and G. Zanetti, "Scaling of hard thermal turbulence in Rayleigh-Bénard convection," J. Fluid Mech. 204, 1 (1989); Ya. G. Sinai and V. Yakhot, "Limiting probability distributions of a passive scalar in a random velocity field," Phys. Rev. Lett. 63, 1962 (1989); H. Chen, S. Chen, and R. H. Kraichnan, "Probability distribution of a stochastically convected scalar field," ibid. 63, 2657 (1989); A. Crisanti, M. Falcioni, G. Paladin, and A. Vulpiani, "Role of Lagrangian chaoticity on the small scale structure of passive scalars in fluids," Physica A 166, 305 (1990); R. Ramshakar and J. Gollub, "Transport by capillary waves, Part II: Scalar dispersion and structure of the concentration field," Phys. Fluids A 3, 1344 (1991); A. Majda, "The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions," ibid. 5, 1963 (1993); B. Shraiman and E. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
15. T. M. Antonsen, Jr. and E. Ott, "Multifractal power spectra of passive scalars convected by chaotic fluid flows," Phys. Rev. A 44, 851 (1991).
16. E. Ott and T. M. Antonsen, Jr. "Fractal measures of passively convected vector fields and scalar gradients in chaotic fluid flows," Phys. Rev. A 39, 3660 (1989);
17. "Chaotic fluid convection and the fractal nature of passive scalar gradients," Phys. Rev. Lett. 61, 2839 (1988).
18. R. Pierrehumbert, "On tracer microstructure in the large-eddy dominated regime," Chaos, Solitons Fractals 4, 1091 (1994).
19. A. M. Obukov, "Structure of the temperature field in turbulent flow," Izv. Akad. Nauk. Geogr. Geofiz. 13, 58 (1949); S. Corrsin, "On the spectrum of isotropic temperature fluctuations in an isotropic turbulence," J. Appl. Phys. 22, 469 (1951); G. K. Batchelor, "Small-scale variation of convected quantities like temperature in a turbulent fluid. Part I," J. Fluid Mech. 5, 113 (1959); R. H. Kraichnan, "Small-scale structure of a scalar field convected by turbulence," Phys. Fluids 11, 945 (1968); R. H. Kraichnan, "Convection of a passive scalar by a quasi-uniform random stirring field," J. Fluid Mech. 64, 737 (1974).
20. A. M. Obukov, "Structure of the temperature field in turbulent flow," Izv. Akad. Nauk. Geogr. Geofiz. 13, 58 (1949); S. Corrsin, "On the spectrum of isotropic temperature fluctuations in an isotropic turbulence," J. Appl. Phys. 22, 469 (1951); G. K. Batchelor, "Small-scale variation of convected quantities like temperature in a turbulent fluid. Part I," J. Fluid Mech. 5, 113 (1959); R. H. Kraichnan, "Small-scale structure of a scalar field convected by turbulence," Phys. Fluids 11, 945 (1968); R. H. Kraichnan, "Convection of a passive scalar by a quasi-uniform random stirring field," J. Fluid Mech. 64, 737 (1974).
21. A. M. Obukov, "Structure of the temperature field in turbulent flow," Izv. Akad. Nauk. Geogr. Geofiz. 13, 58 (1949); S. Corrsin, "On the spectrum of isotropic temperature fluctuations in an isotropic turbulence," J. Appl. Phys. 22, 469 (1951); G. K. Batchelor, "Small-scale variation of convected quantities like temperature in a turbulent fluid. Part I," J. Fluid Mech. 5, 113 (1959); R. H. Kraichnan, "Small-scale structure of a scalar field convected by turbulence," Phys. Fluids 11, 945 (1968); R. H. Kraichnan, "Convection of a passive scalar by a quasi-uniform random stirring field," J. Fluid Mech. 64, 737 (1974).
22. A. M. Obukov, "Structure of the temperature field in turbulent flow," Izv. Akad. Nauk. Geogr. Geofiz. 13, 58 (1949); S. Corrsin, "On the spectrum of isotropic temperature fluctuations in an isotropic turbulence," J. Appl. Phys. 22, 469 (1951); G. K. Batchelor, "Small-scale variation of convected quantities like temperature in a turbulent fluid. Part I," J. Fluid Mech. 5, 113 (1959); R. H. Kraichnan, "Small-scale structure of a scalar field convected by turbulence," Phys. Fluids 11, 945 (1968); R. H. Kraichnan, "Convection of a passive scalar by a quasi-uniform random stirring field," J. Fluid Mech. 64, 737 (1974).
23. A. M. Obukov, "Structure of the temperature field in turbulent flow," Izv. Akad. Nauk. Geogr. Geofiz. 13, 58 (1949); S. Corrsin, "On the spectrum of isotropic temperature fluctuations in an isotropic turbulence," J. Appl. Phys. 22, 469 (1951); G. K. Batchelor, "Small-scale variation of convected quantities like temperature in a turbulent fluid. Part I," J. Fluid Mech. 5, 113 (1959); R. H. Kraichnan, "Small-scale structure of a scalar field convected by turbulence," Phys. Fluids 11, 945 (1968); R. H. Kraichnan, "Convection of a passive scalar by a quasi-uniform random stirring field," J. Fluid Mech. 64, 737 (1974).
24. R. R. Prasad, C. Meneveau, and K. R. Sreenivasan, "Multifractal nature of the dissipation field of passive scalars in fully developed turbulence," Phys. Rev. Lett. 61, 74 (1988). See this paper for additional relevant references on experiments in turbulent fluids.
25. P. Grassberger, R. Badii, and A. Politi, "Scaling laws for invariant measures on hyperbolic and nonhyperbolic attractors," J. Stat. Phys. 51, 135 (1988); H. Hata, T. Horita, H. Mori, T. Morita, and K. Tomita, "Characterization of local structures of chaotic attractors in terms of coarse-grained local expansion rates," Prog. Theor. Phys. 80, 809 (1988); E. Ott, C. Grebogi, and J. A. Yorke, "Theory of first order phase transitions for chaotic attractors of nonlinear dynamical systems," Phys. Lett. 135A, 343 (1989).
26. P. Grassberger, R. Badii, and A. Politi, "Scaling laws for invariant measures on hyperbolic and nonhyperbolic attractors," J. Stat. Phys. 51, 135 (1988); H. Hata, T. Horita, H. Mori, T. Morita, and K. Tomita, "Characterization of local structures of chaotic attractors in terms of coarse-grained local expansion rates," Prog. Theor. Phys. 80, 809 (1988); E. Ott, C. Grebogi, and J. A. Yorke, "Theory of first order phase transitions for chaotic attractors of nonlinear dynamical systems," Phys. Lett. 135A, 343 (1989).
27. P. Grassberger, R. Badii, and A. Politi, "Scaling laws for invariant measures on hyperbolic and nonhyperbolic attractors," J. Stat. Phys. 51, 135 (1988); H. Hata, T. Horita, H. Mori, T. Morita, and K. Tomita, "Characterization of local structures of chaotic attractors in terms of coarse-grained local expansion rates," Prog. Theor. Phys. 80, 809 (1988); E. Ott, C. Grebogi, and J. A. Yorke, "Theory of first order phase transitions for chaotic attractors of nonlinear dynamical systems," Phys. Lett. 135A, 343 (1989).
28. M. A. Sepulveda, R. Badii, and E. Pollak, "Spectral analysis of conservative dynamical systems," Phys. Rev. Lett. 63, 1226 (1989).
29. T. Horita, H. Hata, R. Ishizaki, and H. Mori, "Long-time correlations and expansion rate spectra of chaos in Hamiltonian systems," Prog. Theor. Phys. B 3, 1065 (1990).
30. F. Varosi, T. M. Antonsen, Jr., and E. Ott, "The spectrum of fractal dimensions of passively convected scalar gradients in chaotic fluid flows," Phys. Fluids A 3, 1017 (1991).
31. F. J. Muzzio, P. D. Swanson, and J. M. Ottino, "The statistics of stretching and stirring in chaotic flows," Phys. Fluids A 3, 822 (1991); F. J. Muzzio, C. Meneveau, P. D. Swanson, and J. M. Ottino, "Scaling and multifractal properties of mixing in chaotic fluids," Phys. Fluids A 4, 1439 (1992).
32. F. J. Muzzio, P. D. Swanson, and J. M. Ottino, "The statistics of stretching and stirring in chaotic flows," Phys. Fluids A 3, 822 (1991); F. J. Muzzio, C. Meneveau, P. D. Swanson, and J. M. Ottino, "Scaling and multifractal properties of mixing in chaotic fluids," Phys. Fluids A 4, 1439 (1992).
33. S. Friedlander and M. Vishik, "Dynamo theory, vorticity generation, and exponential stretching," Chaos 1, 198 (1991); A. Lifshitz and E. Hameiri, "Local stability conditions in fluid dynamics," Phys. Fluids A 3, 2644 (1991).
34. S. Friedlander and M. Vishik, "Dynamo theory, vorticity generation, and exponential stretching," Chaos 1, 198 (1991); A. Lifshitz and E. Hameiri, "Local stability conditions in fluid dynamics," Phys. Fluids A 3, 2644 (1991).